Construct Single-Hierarchical P/NBD Model for Online Retail Transaction Data
In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.
1 Load and Construct Datasets
We start by modelling the P/NBD model using our synthetic datasets before we try to model real-life data.
Show code
use_fit_start_date <- as.Date("2009-12-01")
use_fit_end_date <- as.Date("2010-12-01")
use_valid_start_date <- as.Date("2010-12-01")
use_valid_end_date <- as.Date("2011-12-31")1.1 Load Online Retail Data
We now want to load the online retail transaction data.
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customer_cohortdata_tbl <- read_rds("data/onlineretail_cohort_tbl.rds")
customer_cohortdata_tbl |> glimpse()Rows: 5,852
Columns: 5
$ customer_id <chr> "12346", "12347", "12348", "12349", "12350", "12351", …
$ cohort_qtr <chr> "2010 Q1", "2010 Q4", "2010 Q3", "2010 Q2", "2011 Q1",…
$ cohort_ym <chr> "2010 03", "2010 10", "2010 09", "2010 04", "2011 02",…
$ first_tnx_date <date> 2010-03-02, 2010-10-31, 2010-09-27, 2010-04-29, 2011-…
$ total_tnx_count <int> 3, 8, 5, 3, 1, 1, 9, 2, 1, 2, 6, 2, 5, 10, 6, 4, 10, 2…
Show code
customer_transactions_tbl <- read_rds("data/onlineretail_transactions_tbl.rds")
customer_transactions_tbl |> glimpse()Rows: 53,711
Columns: 4
$ tnx_timestamp <dttm> 2009-12-01 07:45:00, 2009-12-01 07:45:59, 2009-12-01 09…
$ invoice_id <chr> "489434", "489435", "489436", "489437", "489438", "48943…
$ customer_id <chr> "13085", "13085", "13078", "15362", "18102", "12682", "1…
$ tnx_amount <dbl> 505.30, 145.80, 630.33, 310.75, 2286.24, 426.30, 50.40, …
1.2 Load Derived Data
Show code
customer_summarystats_tbl <- read_rds("data/onlineretail_customer_summarystats_tbl.rds")
obs_fitdata_tbl <- read_rds("data/onlineretail_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/onlineretail_obs_validdata_tbl.rds")
customer_fit_stats_tbl <- obs_fitdata_tbl |>
rename(x = tnx_count)Finally, we need to set our directories where we save our Stan code and the model outputs.
Show code
stan_modeldir <- "stan_models"
stan_codedir <- "stan_code"2 Fit First Hierarchical Lambda Model
Our first hierarchical model puts a hierarchical prior around the mean of our population \(\lambda\) - lambda_mn.
Once again we use a Gamma prior for it.
2.1 Compile and Fit Stan Model
We now compile this model using CmdStanR.
Show code
pnbd_onehierlambda_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_lambda.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
Show code
stan_modelname <- "pnbd_onlineretail_onehierlambda1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.25,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_onlineretail_onehierlambda1_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_onlineretail_onehierlambda1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_onlineretail_onehierlambda1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_onlineretail_onehierlambda1_stanfit$summary()# A tibble: 13,011 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -6.89e+4 -6.89e+4 7.46e+1 7.27e+1 -6.90e+4 -6.88e+4 1.00 670.
2 lambda_mn 1.68e-1 1.68e-1 3.26e-3 3.28e-3 1.62e-1 1.73e-1 1.00 1414.
3 lambda[1] 2.79e-1 2.73e-1 7.31e-2 7.10e-2 1.72e-1 4.05e-1 1.00 3020.
4 lambda[2] 1.13e-1 7.74e-2 1.22e-1 8.28e-2 4.39e-3 3.55e-1 1.00 2514.
5 lambda[3] 9.67e-2 6.28e-2 1.07e-1 6.53e-2 4.48e-3 3.04e-1 1.00 2014.
6 lambda[4] 7.10e-2 6.49e-2 3.44e-2 3.26e-2 2.48e-2 1.33e-1 1.00 2548.
7 lambda[5] 1.63e-1 1.15e-1 1.61e-1 1.19e-1 9.47e-3 4.91e-1 1.00 2005.
8 lambda[6] 2.35e-1 2.02e-1 1.68e-1 1.48e-1 4.00e-2 5.44e-1 1.00 2504.
9 lambda[7] 1.07e-1 7.08e-2 1.15e-1 7.44e-2 4.59e-3 3.18e-1 1.00 1970.
10 lambda[8] 9.88e-2 5.83e-2 1.11e-1 6.36e-2 4.01e-3 3.24e-1 1.00 2284.
# ℹ 13,001 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_onlineretail_onehierlambda1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
2.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_onlineretail_onehierlambda1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_onlineretail_onehierlambda1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")2.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_onlineretail_onehierlambda1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_onlineretail_onehierlambda1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_onlineretail_onehierlambda1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_onlineretail_onehierlambda1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_valid_simstats_tbl.rds"
2.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
2.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
3 Fit Second Hierarchical Lambda Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
3.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_onlineretail_onehierlambda2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.50,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_onlineretail_onehierlambda2_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_onlineretail_onehierlambda2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_onlineretail_onehierlambda2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_onlineretail_onehierlambda2_stanfit$summary()# A tibble: 13,011 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -6.89e+4 -6.89e+4 7.42e+1 7.47e+1 -6.90e+4 -6.88e+4 1.00 603.
2 lambda_mn 1.68e-1 1.68e-1 3.12e-3 3.18e-3 1.63e-1 1.73e-1 1.00 1548.
3 lambda[1] 2.76e-1 2.68e-1 7.31e-2 7.26e-2 1.71e-1 4.10e-1 1.00 2012.
4 lambda[2] 1.12e-1 7.32e-2 1.23e-1 7.40e-2 5.10e-3 3.53e-1 1.00 1889.
5 lambda[3] 9.88e-2 6.48e-2 1.10e-1 6.77e-2 3.88e-3 3.09e-1 1.00 1681.
6 lambda[4] 6.93e-2 6.39e-2 3.46e-2 3.24e-2 2.44e-2 1.34e-1 1.01 2013.
7 lambda[5] 1.67e-1 1.17e-1 1.63e-1 1.17e-1 9.08e-3 5.12e-1 1.00 1948.
8 lambda[6] 2.29e-1 1.97e-1 1.61e-1 1.39e-1 4.16e-2 5.30e-1 1.00 1636.
9 lambda[7] 1.08e-1 7.23e-2 1.13e-1 7.71e-2 4.74e-3 3.35e-1 1.00 1442.
10 lambda[8] 1.00e-1 6.08e-2 1.17e-1 6.80e-2 3.75e-3 3.28e-1 1.00 1420.
# ℹ 13,001 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_onlineretail_onehierlambda2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
3.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_onlineretail_onehierlambda2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_onlineretail_onehierlambda2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")3.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_onlineretail_onehierlambda2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_onlineretail_onehierlambda2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_onlineretail_onehierlambda2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_onlineretail_onehierlambda2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_valid_simstats_tbl.rds"
3.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehierlambda2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
3.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
4 Fit First Hierarchical Mu Model
We now construct the same hierarchical model but based around mu_mn.
4.1 Compile and Fit Stan Model
We compile this model using CmdStanR.
Show code
pnbd_onehiermu_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_mu.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
Show code
stan_modelname <- "pnbd_onlineretail_onehiermu1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.50,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_onlineretail_onehiermu1_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_onlineretail_onehiermu1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_onlineretail_onehiermu1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_onlineretail_onehiermu1_stanfit$summary()# A tibble: 13,011 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -6.40e+4 -6.40e+4 7.49e+1 7.53e+1 -6.41e+4 -6.39e+4 1.00 596.
2 mu_mn 7.37e-3 7.36e-3 5.31e-4 5.11e-4 6.49e-3 8.28e-3 1.03 114.
3 lambda[1] 2.81e-1 2.74e-1 7.67e-2 7.57e-2 1.70e-1 4.16e-1 1.00 2433.
4 lambda[2] 1.23e-1 8.50e-2 1.26e-1 9.08e-2 5.15e-3 3.62e-1 0.999 1737.
5 lambda[3] 8.22e-2 5.69e-2 8.50e-2 5.99e-2 4.50e-3 2.50e-1 1.00 2036.
6 lambda[4] 7.39e-2 6.75e-2 3.57e-2 3.43e-2 2.71e-2 1.39e-1 1.00 2648.
7 lambda[5] 2.38e-1 1.66e-1 2.39e-1 1.68e-1 1.34e-2 7.10e-1 1.00 1943.
8 lambda[6] 3.01e-1 2.55e-1 2.13e-1 1.89e-1 5.30e-2 7.29e-1 1.01 1698.
9 lambda[7] 1.16e-1 7.84e-2 1.16e-1 7.97e-2 7.19e-3 3.50e-1 1.00 1822.
10 lambda[8] 5.51e-2 2.86e-2 8.17e-2 3.13e-2 2.04e-3 1.94e-1 1.00 1893.
# ℹ 13,001 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_onlineretail_onehiermu1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
4.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_onlineretail_onehiermu1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_onlineretail_onehiermu1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")4.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_onlineretail_onehiermu1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_onlineretail_onehiermu1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_onlineretail_onehiermu1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_onlineretail_onehiermu1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_valid_simstats_tbl.rds"
4.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehiermu1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
4.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
5 Fit Second Hierarchical Mu Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
5.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_onlineretail_onehiermu2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.25,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_onlineretail_onehiermu2_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_onlineretail_onehiermu2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_onlineretail_onehiermu2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_onlineretail_onehiermu2_stanfit$summary()# A tibble: 13,011 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -6.40e+4 -6.40e+4 7.29e+1 7.01e+1 -6.41e+4 -6.39e+4 1.00 649.
2 mu_mn 7.36e-3 7.34e-3 5.43e-4 5.44e-4 6.51e-3 8.25e-3 1.06 65.7
3 lambda[1] 2.79e-1 2.74e-1 7.34e-2 7.13e-2 1.72e-1 4.13e-1 1.01 2153.
4 lambda[2] 1.19e-1 8.22e-2 1.21e-1 9.02e-2 4.73e-3 3.68e-1 1.00 1175.
5 lambda[3] 8.09e-2 5.37e-2 8.96e-2 5.43e-2 4.79e-3 2.45e-1 1.00 1821.
6 lambda[4] 7.02e-2 6.44e-2 3.67e-2 3.51e-2 2.21e-2 1.38e-1 1.00 1298.
7 lambda[5] 2.40e-1 1.66e-1 2.43e-1 1.79e-1 1.07e-2 7.20e-1 1.00 1620.
8 lambda[6] 3.02e-1 2.66e-1 2.02e-1 1.84e-1 5.74e-2 6.83e-1 1.00 1746.
9 lambda[7] 1.13e-1 7.81e-2 1.16e-1 7.99e-2 5.41e-3 3.21e-1 1.00 1477.
10 lambda[8] 5.77e-2 3.11e-2 8.84e-2 3.34e-2 1.87e-3 2.07e-1 1.00 1431.
# ℹ 13,001 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_onlineretail_onehiermu2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
The following parameters had split R-hat greater than 1.05:
mu_mn, beta
Such high values indicate incomplete mixing and biased estimation.
You should consider regularizating your model with additional prior information or a more effective parameterization.
Processing complete.
5.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_onlineretail_onehiermu2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_onlineretail_onehiermu2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")5.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_onlineretail_onehiermu2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_onlineretail_onehiermu2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_onlineretail_onehiermu2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_onlineretail_onehiermu2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_valid_simstats_tbl.rds"
5.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehiermu2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
5.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_onlineretail_onehiermu1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
6 Compare Model Outputs
We have looked at each of the models individually, but it is also worth looking at each of the models as a group.
Show code
calculate_simulation_statistics <- function(file_rds) {
simdata_tbl <- read_rds(file_rds)
multicount_cust_tbl <- simdata_tbl |>
filter(sim_tnx_count > 0) |>
count(draw_id, name = "multicust_count")
totaltnx_data_tbl <- simdata_tbl |>
count(draw_id, wt = sim_tnx_count, name = "simtnx_count")
simstats_tbl <- multicount_cust_tbl |>
inner_join(totaltnx_data_tbl, by = "draw_id")
return(simstats_tbl)
}Show code
obs_fit_customer_count <- obs_fitdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_valid_customer_count <- obs_validdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_fit_total_count <- obs_fitdata_tbl |>
pull(tnx_count) |>
sum()
obs_valid_total_count <- obs_validdata_tbl |>
pull(tnx_count) |>
sum()
obs_stats_tbl <- tribble(
~assess_type, ~name, ~obs_value,
"fit", "multicust_count", obs_fit_customer_count,
"fit", "simtnx_count", obs_fit_total_count,
"valid", "multicust_count", obs_valid_customer_count,
"valid", "simtnx_count", obs_valid_total_count
)
model_assess_tbl <- dir_ls("data", regexp = "pnbd_onlineretail_(one|fixed).*_assess") |>
enframe(name = NULL, value = "file_path") |>
filter(str_detect(file_path, "_assess_model_", negate = TRUE)) |>
mutate(
model_label = str_replace(file_path, "data/pnbd_onlineretail_(.*?)_assess_.*", "\\1"),
assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
sim_data = map(
file_path, calculate_simulation_statistics,
.progress = "calculate_simulation_statistics"
)
)
model_assess_tbl |> glimpse()Rows: 16
Columns: 4
$ file_path <fs::path> "data/pnbd_onlineretail_fixed1_assess_fit_simstats_tb…
$ model_label <chr> "fixed1", "fixed1", "fixed2", "fixed2", "fixed3", "fixed3"…
$ assess_type <chr> "fit", "valid", "fit", "valid", "fit", "valid", "fit", "va…
$ sim_data <list> [<tbl_df[2000 x 3]>], [<tbl_df[2000 x 3]>], [<tbl_df[2000…
Show code
model_assess_summstat_tbl <- model_assess_tbl |>
select(model_label, assess_type, sim_data) |>
unnest(sim_data) |>
pivot_longer(
cols = !c(model_label, assess_type, draw_id)
) |>
group_by(model_label, assess_type, name) |>
summarise(
.groups = "drop",
mean_val = mean(value),
p10 = quantile(value, 0.10),
p25 = quantile(value, 0.25),
p50 = quantile(value, 0.50),
p75 = quantile(value, 0.75),
p90 = quantile(value, 0.90)
)
model_assess_summstat_tbl |> glimpse()Rows: 32
Columns: 9
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed2", "fixed2"…
$ assess_type <chr> "fit", "fit", "valid", "valid", "fit", "fit", "valid", "va…
$ name <chr> "multicust_count", "simtnx_count", "multicust_count", "sim…
$ mean_val <dbl> 2778.159, 15495.754, 1979.650, 23733.976, 2880.884, 11873.…
$ p10 <dbl> 2743.9, 15108.0, 1949.0, 22894.7, 2844.0, 11534.0, 1479.0,…
$ p25 <dbl> 2759.00, 15279.00, 1964.00, 23275.50, 2860.00, 11690.00, 1…
$ p50 <dbl> 2779.0, 15502.0, 1979.0, 23738.5, 2882.0, 11877.0, 1511.5,…
$ p75 <dbl> 2796.00, 15696.25, 1996.00, 24180.25, 2901.25, 12045.25, 1…
$ p90 <dbl> 2814.0, 15896.0, 2012.0, 24536.0, 2918.0, 12203.0, 1543.0,…
Show code
#! echo: TRUE
ggplot(model_assess_summstat_tbl) +
geom_errorbar(
aes(x = model_label, ymin = p10, ymax = p90), width = 0
) +
geom_errorbar(
aes(x = model_label, ymin = p25, ymax = p75), width = 0, linewidth = 3
) +
geom_hline(
aes(yintercept = obs_value),
data = obs_stats_tbl, colour = "red"
) +
scale_y_continuous(labels = label_comma()) +
expand_limits(y = 0) +
facet_wrap(
vars(assess_type, name), scale = "free_y"
) +
labs(
x = "Model",
y = "Count",
title = "Comparison Plot for the Different Models"
) +
theme(
axis.text.x = element_text(angle = 20, vjust = 0.5, size = 8)
)6.1 Write Assessment Data to Disk
We now want to save the assessment data to disk.
Show code
model_assess_tbl |> write_rds("data/assess_data_pnbd_onlineretail_onehier_tbl.rds")7 R Environment
Show code
options(width = 120L)
sessioninfo::session_info()─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
setting value
version R version 4.2.3 (2023-03-15)
os Ubuntu 22.04.2 LTS
system x86_64, linux-gnu
ui X11
language (EN)
collate en_US.UTF-8
ctype en_US.UTF-8
tz Europe/Dublin
date 2023-06-09
pandoc 2.19.2 @ /usr/local/bin/ (via rmarkdown)
─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
package * version date (UTC) lib source
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base64enc 0.1-3 2015-07-28 [1] RSPM (R 4.2.0)
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svUnit 1.0.6 2021-04-19 [1] RSPM (R 4.2.0)
tensorA 0.36.2 2020-11-19 [1] RSPM (R 4.2.0)
threejs 0.3.3 2020-01-21 [1] RSPM (R 4.2.0)
tibble * 3.2.1 2023-03-20 [1] RSPM (R 4.2.0)
tidybayes * 3.0.4 2023-03-14 [1] RSPM (R 4.2.0)
tidyr * 1.3.0 2023-01-24 [1] RSPM (R 4.2.0)
tidyselect 1.2.0 2022-10-10 [1] RSPM (R 4.2.0)
tidyverse * 2.0.0 2023-02-22 [1] RSPM (R 4.2.0)
timechange 0.2.0 2023-01-11 [1] RSPM (R 4.2.0)
tzdb 0.3.0 2022-03-28 [1] RSPM (R 4.2.0)
utf8 1.2.3 2023-01-31 [1] RSPM (R 4.2.0)
vctrs 0.6.2 2023-04-19 [1] RSPM (R 4.2.0)
withr 2.5.0 2022-03-03 [1] RSPM (R 4.2.0)
xfun 0.38 2023-03-24 [1] RSPM (R 4.2.0)
xtable 1.8-4 2019-04-21 [1] RSPM (R 4.2.0)
xts 0.13.1 2023-04-16 [1] RSPM (R 4.2.0)
yaml 2.3.7 2023-01-23 [1] RSPM (R 4.2.0)
zoo 1.8-12 2023-04-13 [1] RSPM (R 4.2.0)
[1] /usr/local/lib/R/site-library
[2] /usr/local/lib/R/library
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options(width = 80L)